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arXiv · 1210.7043

Empty Monochromatic Simplices

Oswin Aichholzer·Ruy Fabila-Monroy·Thomas Hackl·Clemens Huemer·Jorge Urrutia
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Abstract

Let $S$ be a $k$-colored (finite) set of $n$ points in $\mathbb{R}^d$, $d\geq 3$, in general position, that is, no {$(d + 1)$} points of $S$ lie in a common $(d - 1)$}-dimensional hyperplane. We count the number of empty monochromatic $d$-simplices determined by $S$, that is, simplices which have only points from one color class of $S$ as vertices and no points of $S$ in their interior. For $3 \leq k \leq d$ we provide a lower bound of $\Omega(n^{d-k+1+2^{-d}})$ and strengthen this to $\Omega(n^{d-2/3})$ for $k=2$. On the way we provide various results on triangulations of point sets in $\mathbb{R}^d$. In particular, for any constant dimension $d\geq3$, we prove that every set of $n$ points ($n$ sufficiently large), in general position in $\mathbb{R}^d$, admits a triangulation with at least $dn+\Omega(\log n)$ simplices.

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Oswin Aichholzer, Ruy Fabila-Monroy, Thomas Hackl, Clemens Huemer, Jorge Urrutia. 2012-10-26. Empty Monochromatic Simplices. https://arxiv.org/abs/1210.7043

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